3D shapes and nets - KS3 Maths

Key points

A series of eight images. Each pair of images shows a matching three dimensional shape and its net. — Image caption,
3D shapes can be made from 2D shapes called nets.

A shape can be made from a shape called a .
The two-dimensional shapes that form a net can be arranged in different ways for a particular 3D shape. The relationship of faces, and edges must remain the same. There are eleven different nets that will form a cube.
An is used as a guide for drawing a 3D shape on paper.

Video

Watch the video to learn how 3D shapes, such as a cube, can be created from nets, an arrangement of 2D shapes.

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How to draw the net of a cube and a cuboid

To draw a net of a or a , imagine the shape is being unfolded. Each face is a or square.
- Draw the , which has the same dimensions as the top face.
- Draw the faces that are attached to the base face.
- Draw the top face attached to an edge of the front or back, or left or right face.
- The top face must not touch the base face at all (edge or vertex). The top and base faces do not touch on the 3D shape, this is also true on the net.
- The net of a cube is called a hexomino. Hex is a prefix in Greek meaning 'six'. The net is made up of six congruent squares joined together along their edges. There are eleven different ways of arranging the square for a correct net of a cube.

Examples

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A net is a 2D shape that folds to make up a 3D shape. The net of a dodecahedron (a platonic solid with twelve pentagonal faces) is made up of 12 regular pentagons.
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Draw the net of a cube.
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A cube has six faces. The net of a cube is made up of six squares. First draw the square base face. Then draw the faces that are attached to the base face. These are the front and back face and the left and right faces.
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The sixth square on the net is the top face. There are several possible positions for the top face. The top face may be attached to an edge of the front or back or left or right face. The top face of the cube does not touch the base face. On the net the top face must not touch an edge or a vertex of the base face.
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This net is in the shape of a cross. The opposite faces of the cube are not next to each other on the net.
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There are eleven possible nets that will form a cube.
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Draw the net of a 6 cm by 2 cm by 4 cm cuboid.
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First draw a rectangle for the base face. The base measures 6 cm by 4 cm. Then draw the faces that are attached to the base face. The left and right faces measure 4 cm by 2 cm. The front and back faces measure 6 cm by 2 cm.
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The final rectangle on the net is the top face. There are several possible positions for the top face. The top face may be attached to an edge of the front or back or left or right face. The top face must not touch an edge or a vertex of the base face. The top face is congruent to the base face, it measures 6 cm by 4 cm. The net is complete.
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The three rectangles used to make the cuboid net give the dimensions of the cuboid. The rectangles are 7 cm by 8 cm, 7 cm by 3 cm and 3 cm by 8 cm. The dimensions of the cuboid for this net are 7 cm, 8 cm and 3 cm. The length, width and height can be any combination of these measurements.

How to draw the nets of a prism and a pyramid

To draw the net of a triangular :
- The net is made up of three rectangles and two triangles.
- For a that is an equilateral triangle, the rectangles are .
- For a cross-section that is an , two of the rectangles are congruent.
- For a cross-section that is a , the rectangles are all different.
- The rectangles are drawn next to each other, joined along the dimension that is the length of the prism.
- A triangle is drawn at each end of one of the rectangles, making sure that when folded the edge lengths match exactly.
To draw the net of a prism with a regular cross-section:
- The net is made up of congruent rectangles and two polygon ends.
- The number of rectangles is equal to the number of sides of the polygon. One dimension of a rectangle is the length of the prism, the other is the side-length of the polygon cross-section.
- The rectangles are drawn next to each other, joined along the dimension that is the length of the prism.
- The polygon is drawn at each end of one of the rectangles.
To draw the net of a :
- The net is made up of the base polygon and congruent isosceles triangles attached to each side of the base.
- The measurements given are the side-length of the polygon and the height of each triangle. The height of the triangle is the of the pyramid.
- Draw the base shape.
- On each side of the base draw an isosceles triangle.

Examples

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Draw the net of a triangular prism.
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The cross-section of the prism is a right-angled triangle with side-lengths 3 cm, 4 cm and 5 cm. The prism is 6 cm long. The net is made up of two right-angled triangles and three rectangles. The dimensions of the rectangles are 3 cm by 6 cm, 4 cm by 6 cm, and 5 cm by 6 cm.
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Draw the three rectangles next to each other joined along the dimension that is the length of the prism (6 cm).
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Draw a right-angled triangle at opposite ends of a rectangle. The net is complete.
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The net may be drawn in more than one way.
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Sketch the net of a hexagonal prism.
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There are no measurements on the 3D shape. The sketch will show the shapes used to form the net of the hexagonal prism. The cross-section of the prism is a regular hexagon. The net is made up of two hexagons and six rectangles.
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Draw the six rectangles next to each other joined along the dimension that is the length of the prism (5 cm) and draw a hexagon at opposite ends of one of the rectangles. The net is complete.
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Draw the net of a square-based pyramid. The square has sides-lengths of 4 cm, the slant height of the pyramid is 6 cm.
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The net is made up of a square and four congruent isosceles triangles. Draw the square 4 cm by 4 cm base and attach the four congruent isosceles triangles with base 4 cm and height 6 cm on each side of the square. The net is complete.

Question

A cuboid is drawn on isometric paper. What are the dimensions of the cuboid?

An image of a cuboid drawn on triangular spotty paper. It has a length measuring eight centimetres, width measuring four centimetres and a height measuring six centimetres.

How to draw a 2D representation of a 3D shape

A 3D shape may be drawn on plain paper as a sketch or on an .

To draw any 3D shape with a constant on plain paper:
- Use faint lines when drawing, so that they can be made heavier or lighter as needed when complete.
- Draw the shape of the cross-section.
- Draw the same shape higher and to the right of the original shape.
- Join the corresponding vertices.
- For a solid 3D shape, rub out the edges that would be hidden from view.
- For a skeleton 3D shape, make the hidden edges dotted lines.
To draw a on an isometric grid:
- First make sure the paper is the right way round. The correct orientation is when the longest line is vertical.
- Pick a dot, draw a vertical line for the height of the cuboid and draw to the left and up, and to the right and up for the dimensions of the base of the cuboid.
- Draw vertical lines at the ends of the oblique lines.
- Draw the top face of the cuboid with two pairs of parallel lines.

Examples

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Sketch a prism on plain paper.
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Draw the shape of the cross-section. The shape of the cross-section is a trapezium. Use faint lines when drawing so that they can be made heavier or lighter as needed when complete.
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Draw the same trapezium higher and to the right of the original shape and then join the corresponding vertices.
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For a solid 3D shape, rub out the edges that would be hidden from view. For a skeleton 3D shape make the hidden edges dotted lines.
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Drawing the second trapezium in different positions gives a different result.
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Draw a 3 cm by 4 cm by 5 cm cuboid on isometric paper.
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First make sure the isometric grid is the right way round. Pick a dot and draw straight lines to the six nearest dots. The correct orientation is when the longest line is vertical. When a 3D shape is drawn on the grid, the lines will be drawn in the directions shown by the diagram used to orientate the paper. No horizontal lines are drawn. The dimensions of the cuboid are in centimetres. Treat the shortest gaps between the dots as 1 cm.
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Pick a dot and draw a vertical line to represent 5 cm for the height of the cuboid. Draw oblique lines representing 3 cm to the left and up and 4 cm to the right and up for the dimensions of the base of the cuboid.
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Draw a vertical line at the end of each oblique line to show the 5 cm edges of the cuboid.
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Complete the cuboid by drawing the top face of the cuboid with two parallel 3 cm lines and two parallel 4 cm lines.

Question

On plain paper draw a copy of this 3D shape.

An image of a prism. The cross section is an L shape.

Practise your understanding of 3D shapes and nets

Quiz

Practise applying your knowledge of 3D shapes and nets with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

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Flat cardboard is punched with nets for folding up into boxes.

Nets can be used in any organisation that packs goods. Flat cardboard is punched with nets that include flaps for fixing or with additional flaps for folding up into boxes.

The machinery used will punch out the net and fold it up into the box or container ready to be filled.

The design of the net is such that card waste is kept to a minimum and the fixing together of the net is straight forward. Some machines will staple the boxes, others will tape and others will fold without any other fixing.